Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.0917 - 0.995i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.26 − 2.18i)2-s + (−1.5 − 0.866i)3-s + (−1.17 + 2.03i)4-s + (−1.93 + 1.11i)5-s + 4.36i·6-s + (−6.18 + 3.28i)7-s − 4.15·8-s + (1.5 + 2.59i)9-s + (4.88 + 2.81i)10-s + (4.36 − 7.55i)11-s + (3.52 − 2.03i)12-s + 21.5i·13-s + (14.9 + 9.34i)14-s + 3.87·15-s + (9.93 + 17.2i)16-s + (−18.7 − 10.8i)17-s + ⋯
L(s)  = 1  + (−0.630 − 1.09i)2-s + (−0.5 − 0.288i)3-s + (−0.294 + 0.509i)4-s + (−0.387 + 0.223i)5-s + 0.727i·6-s + (−0.882 + 0.469i)7-s − 0.519·8-s + (0.166 + 0.288i)9-s + (0.488 + 0.281i)10-s + (0.396 − 0.686i)11-s + (0.294 − 0.169i)12-s + 1.65i·13-s + (1.06 + 0.667i)14-s + 0.258·15-s + (0.621 + 1.07i)16-s + (−1.10 − 0.638i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0917 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0917 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.0917 - 0.995i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.0917 - 0.995i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0122309 + 0.0134097i\)
\(L(\frac12)\)  \(\approx\)  \(0.0122309 + 0.0134097i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 + (1.93 - 1.11i)T \)
7 \( 1 + (6.18 - 3.28i)T \)
good2 \( 1 + (1.26 + 2.18i)T + (-2 + 3.46i)T^{2} \)
11 \( 1 + (-4.36 + 7.55i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 21.5iT - 169T^{2} \)
17 \( 1 + (18.7 + 10.8i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (2.71 - 1.56i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (2.05 + 3.55i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 50.8T + 841T^{2} \)
31 \( 1 + (33.9 + 19.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (26.4 + 45.8i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 36.8iT - 1.68e3T^{2} \)
43 \( 1 - 17.6T + 1.84e3T^{2} \)
47 \( 1 + (3.49 - 2.01i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (2.22 - 3.85i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-81.5 - 47.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (63.3 - 36.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-50.2 + 87.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 56.6T + 5.04e3T^{2} \)
73 \( 1 + (-64.8 - 37.4i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (14.4 + 25.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 21.1iT - 6.88e3T^{2} \)
89 \( 1 + (-63.1 + 36.4i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 73.7iT - 9.40e3T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.42435542079526177555092145437, −12.35754500395738099289906318623, −11.46035139411995110914600961624, −10.98066745270341472498320334754, −9.465675717527968707688361696926, −8.916620020811600285259845304784, −7.01838576805479431220737525958, −5.97644962425724822216300478228, −3.82545665102140960518050881921, −2.15252034120205693244089681994, 0.01654616929462408343970243807, 3.66007775418691444447946620530, 5.43251288816893698292694203459, 6.63043722026117820367851487687, 7.52855516410889689575835812396, 8.775474487357503964025972416971, 9.822248004568984652413925477321, 10.89669158783536894064728983822, 12.36241951764858681091102365110, 13.09950082061971681404503144161

Graph of the $Z$-function along the critical line